Probability Two Randomly Started Flower Bloom Periods Overlap

Two kinds of flowers will each begin blooming on a random day within the next $D$ days, independently of each other. Once it starts, the first kind stays in bloom for $d_1$ consecutive days and the second for $d_2$ consecutive days. What is the probability that at some moment both flowers are in bloom at the same time?

Assume each flower's start day is uniform over days 1 through $D$, and a flower that starts on day $s$ is in bloom on days $s$ through $s + d - 1$, even if that runs past day $D$. Two flowers that are both in bloom on some common day count as overlapping. Derive an expression in $D$, $d_1$ and $d_2$, and explain how the answer changes if start times are continuous instead of whole days. The specific values are generated randomly.

Constraints and Clarifications

• $D$, $d_1$ and $d_2$ are positive integers, and $d_1$ or $d_2$ may exceed $D$.
• The two start days are independent, and each is uniform on ${1, \dots, D}$.

Clarifying Questions

• Is the start day a whole day, or a continuous time within the $D$-day window?
• Must each bloom period finish within the $D$ days, or may it run past the end of the window?
• Do two bloom periods that only touch at a boundary, one ending as the other begins, count as overlapping?